Lenny T. Evans MATH 534 Homework 2 February 4, 2009
7.5.1
Each permutation of n elements in cfw_1, . . . , n = Jn , or every element in the symmetric group Sn , is the product of transpositions. Proof: We will prove this by induction on n. Base case (n =
Lenny T. Evans MATH 534 Homework 3 February 11, 2010
7.7.4
If G is the group of nonzero elements of Z/pZ under for a prime p, then for a G, a 0 (mod p) implies ap1 1 (mod p). Proof: We see that G = cfw_, , ., p 1, which is p 1 elements. 12 Clearly G does
Lenny T. Evans MATH 534 Homework 4 March 4, 2010
Lemmas
Lemma 1: In a general ring R, for all x R it is true that (e)x = x(e) = x. Proof: By distributivity, (e)x+x = (e+e)x = 0, so (e)x is the additive inverse of x, usually denoted x, and x(e) + x = x(e +
Lenny T. Evans MATH 534 Homework 5 March 18, 2010
3.2.1
A eld has no ideal other than the zero and unit ideals. Proof: The zero ideal is obviously an ideal in the eld. Let I be an ideal in the eld that is not the zero ideal. For any nonzero b I (which exi